What will be the equation of directrix of translated parabola $x^{2} + 13 = 4 y + 6 x$ ?

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Solution

Given, the equation of translated parabola $x^{2} + 13 = 4 y + 6 x$ .
On reaaranging we get,
$\Rightarrow x^{2} - 6 x + 9 = 4 y - 4$
$\Rightarrow (x - 3)^{2} = 4 (y - 1)$
So, the vertex of this parabola is $(3, 1)$ and $a = 1.$
Now, the equation of directrix of translated parabola is give by $y = k - a$ i.e. $y = 1 - 1 = 0.$
Hence, the equation of directrix for the given translated parabola is $y = 0.$

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